CBS and Time Warner Cable have extended the deadline to reach a new distribution deal for the Eye’s 13 owned-and-operated stations and cablers including Showtime by three hours, until 8 p.m. Eastern Monday — a sign that the two sides may be close to resolving the dispute.

UPDATE (8:05 p.m. ET): TW Cable said it agreed to an extension until 9 p.m., while CBS said the companies “extended their current deal again into the evening while the companies continue to negotiate.”

UPDATE #2 (9:03 p.m. ET):Time Warner Cable said it agreed to an extension until 10 p.m.

UPDATE #3 (9:54 p.m. ET): The talks have been extended through 11 p.m., according to TWC.

UPDATE #4 (10:55 p.m. ET): Now the deadline is midnight Eastern.

The two sides are locked in a dispute over how much Time Warner Cable will pay CBS for retransmission rights to the Eye’s owned-and-operated stations, including WCBS in New York, KCBS in Los Angeles and KTVT in Dallas.

Also affected in the CBS-Time Warner Cable dispute are carriage of CBS cable channels including Showtime Networks and Smithsonian Channel.

The dispute erupted into public-relations battle July 18, when CBS launched an ad campaign spanning TV, radio, print and online (at KeepCBS.com) accusing TW Cable of holding the local stations “hostage.” The companies’ 2008 pact expired June 30, and was subsequently extended through Monday as the sides try to work out a deal.

Time Warner Cable has claimed CBS is demanding more than 600% above what the cable operator pays in retrans fees to independent CBS affiliates in other parts of the U.S. — an “unprecedented” premium, according to the TWC spin. The cable company has set up a microsite at TWCConversations.com with its talking points on the dispute and has run TV spots and print ads.

CBS chief Les Moonves, in a memo to CBS employees last week that was distributed to the media, said Time Warner Cable’s payments to the Eye are “out of whack.”

“It’s not like Time Warner Cable doesn’t have the money,” Moonves said in the memo. “Cable is a very, very profitable business, and Time Warner Cable can certainly afford to pay CBS a fair rate for our programming without passing any added cost on to its customers.”